Connected and Disconnected Topological Spaces

# Connected and Disconnected Topological Spaces

One very important characteristic of a topological space is the concept of connectedness and disconnectedness. We define these terms below.

 Definition: Let $X$ be a topological space. Then $X$ is said to be Disconnected if there exists open sets $A, B \subset X$ such that $A, B \neq \emptyset$, $A \cap B = \emptyset$, and $X = A \cup B$ and the pair $\{ A, B \}$ is called a Separation of $X$. If $X$ is not disconnected then we say that $X$ is Connected. For example, consider the topological space $\mathbb{R}$ with the usual topology formed from open intervals. Consider the topological subspace $\mathbb{Q}$. We claim that $\mathbb{Q}$ is hence a disconnected topological space.

Consider the irrational number $\sqrt{2}$ and let $A$ and $B$ be subsets of $\mathbb{Q}$ defined by:

(1)
\begin{align} \quad A = \{ q \in \mathbb{Q} : q < \sqrt{2} \} \end{align}
(2)
\begin{align} \quad B = \{ q \in \mathbb{Q} : q > \sqrt{2} \} \end{align}

We claim that $\{ A, B \}$ is a separation of $\mathbb{Q}$. First note that $A$ and $B$ are open in $\mathbb{Q}$ with the subspace topology since $A = \mathbb{Q} \cap (-\infty, \sqrt{2})$ and $B = \mathbb{Q} \cap (\sqrt{2}, \infty)$ and $(-\infty, \sqrt{2})$ and $(\sqrt{2}, \infty)$ are open sets in $\mathbb{R}$ with the usual topology.

Furthermore, $A$ and $B$ are nonempty sets since $0 \in A$ and $2 \in B$, for example. We also see that $A \cap B = \emptyset$ since there exists no rational number that is simultaneously less than $\sqrt{2}$ and greater than $\sqrt{2}$. Also, it's not hard to see that:

(3)
\begin{align} \quad \mathbb{Q} = A \cup B \end{align}

Thus $\{ A, B \}$ is a separation of the topological space $\mathbb{Q}$ and so $\mathbb{Q}$ is a disconnected topological spaces.

Note that in general it is much easier to show that a topological space is disconnected than it is to show that a topological space is connected.

If we consider the topological space $\mathbb{R}$ with the usual topology as mentioned above then $\mathbb{R}$ is actually a connected topological space. Showing this (for the time being) is rather cumbersome though. We will develop a stronger type of connectedness later on and prove this as a very simple consequence later on.